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Theorem srglz 16735
Description: The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.)
Hypotheses
Ref Expression
srgz.b  |-  B  =  ( Base `  R
)
srgz.t  |-  .x.  =  ( .r `  R )
srgz.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
srglz  |-  ( ( R  e. SRing  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  .0.  )

Proof of Theorem srglz
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 srgz.b . . . . . . . 8  |-  B  =  ( Base `  R
)
2 eqid 2451 . . . . . . . 8  |-  (mulGrp `  R )  =  (mulGrp `  R )
3 eqid 2451 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
4 srgz.t . . . . . . . 8  |-  .x.  =  ( .r `  R )
5 srgz.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
61, 2, 3, 4, 5issrg 16716 . . . . . . 7  |-  ( R  e. SRing 
<->  ( R  e. CMnd  /\  (mulGrp `  R )  e. 
Mnd  /\  A. x  e.  B  ( A. y  e.  B  A. z  e.  B  (
( x  .x.  (
y ( +g  `  R
) z ) )  =  ( ( x 
.x.  y ) ( +g  `  R ) ( x  .x.  z
) )  /\  (
( x ( +g  `  R ) y ) 
.x.  z )  =  ( ( x  .x.  z ) ( +g  `  R ) ( y 
.x.  z ) ) )  /\  ( (  .0.  .x.  x )  =  .0.  /\  ( x 
.x.  .0.  )  =  .0.  ) ) ) )
76simp3bi 1005 . . . . . 6  |-  ( R  e. SRing  ->  A. x  e.  B  ( A. y  e.  B  A. z  e.  B  ( ( x  .x.  ( y ( +g  `  R ) z ) )  =  ( ( x  .x.  y ) ( +g  `  R
) ( x  .x.  z ) )  /\  ( ( x ( +g  `  R ) y )  .x.  z
)  =  ( ( x  .x.  z ) ( +g  `  R
) ( y  .x.  z ) ) )  /\  ( (  .0. 
.x.  x )  =  .0.  /\  ( x 
.x.  .0.  )  =  .0.  ) ) )
87r19.21bi 2912 . . . . 5  |-  ( ( R  e. SRing  /\  x  e.  B )  ->  ( A. y  e.  B  A. z  e.  B  ( ( x  .x.  ( y ( +g  `  R ) z ) )  =  ( ( x  .x.  y ) ( +g  `  R
) ( x  .x.  z ) )  /\  ( ( x ( +g  `  R ) y )  .x.  z
)  =  ( ( x  .x.  z ) ( +g  `  R
) ( y  .x.  z ) ) )  /\  ( (  .0. 
.x.  x )  =  .0.  /\  ( x 
.x.  .0.  )  =  .0.  ) ) )
98simprd 463 . . . 4  |-  ( ( R  e. SRing  /\  x  e.  B )  ->  (
(  .0.  .x.  x
)  =  .0.  /\  ( x  .x.  .0.  )  =  .0.  ) )
109simpld 459 . . 3  |-  ( ( R  e. SRing  /\  x  e.  B )  ->  (  .0.  .x.  x )  =  .0.  )
1110ralrimiva 2822 . 2  |-  ( R  e. SRing  ->  A. x  e.  B  (  .0.  .x.  x )  =  .0.  )
12 oveq2 6200 . . . 4  |-  ( x  =  X  ->  (  .0.  .x.  x )  =  (  .0.  .x.  X
) )
1312eqeq1d 2453 . . 3  |-  ( x  =  X  ->  (
(  .0.  .x.  x
)  =  .0.  <->  (  .0.  .x. 
X )  =  .0.  ) )
1413rspcv 3167 . 2  |-  ( X  e.  B  ->  ( A. x  e.  B  (  .0.  .x.  x )  =  .0.  ->  (  .0.  .x. 
X )  =  .0.  ) )
1511, 14mpan9 469 1  |-  ( ( R  e. SRing  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   ` cfv 5518  (class class class)co 6192   Basecbs 14278   +g cplusg 14342   .rcmulr 14343   0gc0g 14482   Mndcmnd 15513  CMndccmn 16383  mulGrpcmgp 16698  SRingcsrg 16714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4521
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-iota 5481  df-fv 5526  df-ov 6195  df-srg 16715
This theorem is referenced by:  srgmulgass  16737  srgrmhm  16742
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